Ellipsoid shape parameters

Usage

ell_volume(s)

ell_lode(s)

ell_nadai(s)

ell_jelinek(s)

ell_flinn(s)

ell_size_invariant(s)

ell_strain_invariant(s)

ell_shape_invariant(s)

Arguments

s: numeric. 3-element vector giving the ellipsoid's semi-axis lengths, in any order (can be the square-roots of eigenvalues).

Value

positive numeric

Details

$$e_i = \log s_i$$ with $s1 \geq s2 \geq s3$ the semi-axis lengths of the ellipsoid.

Lode's shape parameter: $$\nu = \frac{2 e_2 - e_1 - e_3}{e_1-e_3}$$ with $e_1 \geq e_2 \geq e_3$. Note that $\nu$ is undefined for spheres, but we arbitrarily declare $\nu=0$ for them. Otherwise $-1 \geq \nu \geq 1$. $\nu=-1$ for prolate spheroids and $\nu=1$ for oblate spheroids.

Octahedral shear strain $e_s$ (Nadai 1963): $$e_s = \sqrt{\frac{(e_1 - e_2)^2 + (e_2 - e_3)^2 + (e_1 - e_3)^2 }{3}}$$

Strain symmetry (Flinn 1963): $$k = \frac{s_1/s_2 - 1}{s_2/s_3 - 1}$$

and strain intensity (Flinn 1963): $$d = \sqrt{(s_1/s_2 - 1)^2 + (s_2/s_3 - 1)^2}$$

Jelinek (1981)'s $P_j$ parameter: $$P_j = e^{\sqrt{2 \vec{v}\cdot \vec{v}}}$$ with $\vec{v} = e_i - \frac{\sum e_i}{3}$

References

Flinn, Derek.(1963): "On the statistical analysis of fabric diagrams." Geological Journal 3.2: 247-253.

Lode, Walter (1926): "Versuche über den Einfluß der mittleren Hauptspannung auf das Fließen der Metalle Eisen, Kupfer und Nickel“ ("Experiments on the influence of the mean principal stress on the flow of the metals iron, copper and nickel"], Zeitschrift für Physik, vol. 36 (November), pp. 913–939, DOI: 10.1007/BF01400222

Nadai, A., and Hodge, P. G., Jr. (1963): "Theory of Flow and Fracture of Solids, vol. II." ASME. J. Appl. Mech. December 1963; 30(4): 640. https://doi.org/10.1115/1.3636654

Jelinek, Vit. "Characterization of the magnetic fabric of rocks." Tectonophysics 79.3-4 (1981): T63-T67.

Examples

# Generate some random data
set.seed(1)
dat <- rvmf(100, k = 20)
s <- principal_stretch(dat)

# Volume of ellipsoid
ell_volume(s)
#> [1] 189.6644

#  Size-related tensor invariant of ellipsoids
ell_size_invariant(s)
#> [1] 3.812844

# Strain-related tensor invariant of ellipsoids
ell_strain_invariant(s)
#> [1] 4.122989

# Shape-related tensor invariant of ellipsoids
ell_shape_invariant(s)
#> [1] 1.370287

# Lode's shape parameter
ell_lode(s)
#> [1] -0.9524619

# Nadai's octahedral shear strain
ell_nadai(s)
#> [1] 1.202446

# Jelinek Pj parameter
ell_jelinek(s)
#> [1] 5.476767

# Flinn's intensity and symmetry parameters
ell_flinn(s)
#> $k
#> [1] 91.0627
#> 
#> $d
#> [1] 3.283228
#>